Theorem eqelssd 3175

Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
eqelssd.1	|- ( ph -> A C_ B )
eqelssd.2	|- ( ( ph /\ x e. B ) -> x e. A )

Assertion

Ref	Expression
eqelssd	|- ( ph -> A = B )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem eqelssd

Step	Hyp	Ref	Expression
1		eqelssd.1	. 2 |- ( ph -> A C_ B )
2		eqelssd.2	. . . 4 |- ( ( ph /\ x e. B ) -> x e. A )
3	2	ex 115	. . 3 |- ( ph -> ( x e. B -> x e. A ) )
4	3	ssrdv 3162	. 2 |- ( ph -> B C_ A )
5	1, 4	eqssd 3173	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   C_ wss 3130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3136  df-ss 3143

This theorem is referenced by:  fiuni  6977  ennnfonelemrn  12420  ennnfonelemdm  12421  unirnblps  13925  unirnbl  13926  dvidlemap  14163  dviaddf  14172  dvimulf  14173  dvcj  14176  dvrecap  14180